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The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5

  • Zoltán Szabó
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Abstract

We determine the vanishing cycles of Xiao's genus-two fibrations Xd→ P1, d=3,4,5, whose fibers admit degree-d maps to a fixed elliptic curve E. We tile the base P1 by triangles and realize the surface Xd as a degree-d branched cover of P1× E. The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For d=4,5, these give new positive factorizations of the identity in the genus-two mapping class group, of types (6,7) and (12,19).

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Cite

BibTeXRIS
BibTeX
@misc{szabo2026monodromy,
  title = {{The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5}},
  author = {Zoltán Szabó},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34042},
  eprint = {2609.34042},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34042},
}
RIS
TY  - UNPB
TI  - The monodromy of Xiao's genus-two fibrations in degrees 3, 4 and 5
AU  - Zoltán Szabó
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34042
UR  - https://arxiv.org/abs/2609.34042
AB  - We determine the vanishing cycles of Xiao's genus-two fibrations X_d→ P^1, d=3,4,5, whose fibers admit degree-d maps to a fixed elliptic curve E. We tile the base P^1 by triangles and realize the surface X_d as a degree-d branched cover of P^1× E. The covering description gives an explicit algorithm for the vanishing cycles. A vanishing path is recorded by the sides it crosses in the tiling; each crossing changes a labelled picture of the fiber by a local hexagon move, and the vanishing cycle is read from the terminal picture. We obtain the $7$, $13$, and $31$ vanishing cycles as explicit curves in a marked reference fiber. For d=4,5, these give new positive factorizations of the identity in the genus-two mapping class group, of types (6,7) and (12,19).
ER  -